Implicit Implementation of Material Point Method for Simulation of Incompressible Problems in Geomechanics

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1 Implicit Implementation of Material Point Method for Simulation of Incompressible Problems in Geomechanics Seyedfarzad Fatemizadeh M.Sc., Institute of Geotechnics (IGS), University of Stuttgart The Finite Element Method (FEM) is becoming a standard tool in engineering science, although its shortcoming in large deformation analysis is apparent. Mesh distortions are common in this area and the remedies like remeshing are computationally expensive. FEM also suffers from locking phenomenon that happens when using low order linear elements in incompressible or nearly incompressible material behaviours (large values for bulk modulus). In this study Material Point Method (MPM) as a powerful tool to simulate large deformatioroblems is presented. MPM discretizes the domain with material points that move through a background mesh. This method is the application of the Particle in Cell Method (PIC) from fluid to solid mechanics and is classified as a meshless method. Most of the MPM applications are formulated with explicit integration schemes. These schemes are conditionally stable and attention should be paid on the correct choice of time step size and computation of the critical time step. However, implicit schemes are unconditionally stable and bigger time steps can be chosen. Here a brief introduction on the governing equations of MPM will be presented. It will be followed by an overview on the Implicit Meshless Dynamic Analysis algorithm as a choice of implicit implementation of MPM. This algorithm is used together with a relaxation method to reach the quasi static solution. As an example the slope stability problem is solved with the developed algorithm using the low order triangular elements and the results are compared with other publications and case studies of slope failure. In this example special attention is paid to overcome the locking problem by averaging the volumetric strains. 1. Introduction In the last decades Finite-Element-Method (FEM) has become a standard tool for analysis of different aspects in wide range of mechanical problems. In the field of geotechnical engineering, simulation of large deformations become more sensible like the cases of slope stability or pile driving. Despite the numerous applications of FEM, especially in the field of solid mechanics, this method is not able to model extremely large deformations. In simulation of large deformations, FEM suffers from what is called the mesh distortion. By using the updated Lagrangian formulation to overcome this shortcoming, errors are introduced during the mapping process (Więckowski, [18]) and remeshing is also computationally expensive. To be able to simulate large deformations, mesh free methods like the Material Point Method (MPM) are developed. MPM uses two different discretizations, one is the space discretization based on the background fixed mesh, and the other one is the material discretization based on the material points or particles. During the

2 simulation, material points that carry all the physical information (strains, stresses, and etc.) can move through the background mesh. With this property extremely large deformations can be modeled. MPM is capable of handling history-dependent material behavior. Background mesh or the Eulerian mesh is used only to solve the governing equations and carries no permanent data. MPM working procedure consists of three phases: first the initializatiohase where all the data (that are already on the particles) are mapped from the particles to the background mesh using the nodal shape functions. Next is the Lagrangiahase where the equations of motion are solved. At last is the convective phase where the updated nodal values are mapped back to the particles. The positions of particles are updated and finally background mesh is brought back to its original position (Fig. 1). The early development of MPM is done by Harlow [9]. He simulated the fluid flow by material points moving through a fixed mesh (what is called the Particle in Cell method). For the first time Sulsky et al. [17] applied this method to solid mechanics and called it Material Point Method (Sulsky et al. [16]). Bardenhagen et al. [1] added a frictional contact algorithm to this method. The first quasi static formulation of MPM was done by Beuth et al. [2]. Since the first formulation of MPM, this method is applied on different applications that show the capability of it to simulate large deformations. Most of this applications use the explicit integration scheme which is conditionally stable and the time step is restricted by the Courant- Friedrichs-Lewy (CFL) condition. If the interest is in the low frequency motions and not the elastic wave motions, one can use the unconditionally stable implicit integration schemes with larger time steps. Cummins and Brackbill [5], Guilkey and Weiss [8], Sulsky and Kaul [15], and Stolle et al. [14] have presented examples of formulation and application of implicit integration schemes in MPM. The layout of the present study is as follows: in section 2 weak formulation and space integration of equilibrium equation in MPM is presented. Section 3 describes the explicit time integration scheme. In section 4 implicit integration scheme based on the method suggested by Stolle et al. [14] is presented. Section 5 presents the method used in this study to overcome locking phenomena that appears in low order elements. In section 6 a slope stability example is solved using both explicit and implicit integration schemes and the results are compared. This is followed by the conclusion in chapter Weak formulation and space integration of equilibrium equation Starting point is the Cauchy form of conservation of linear momentum [11] ρu = σ + ρg and t = σ n (1) Fig. 1 MPM working procedure where ρ is the material density, u is the displacement, a superposed dot declares differentiation with respect to time, σ is the

3 Cauchy stress tensor and g is the gravitational acceleration vector. The surface traction acting on the external boundary is denoted by t and n is the outward unit normal of the boundary. Applying the virtual work principle on a domain of volume surrounded by boundary S yields δu T ρu d = δε T σd + δu T ρgd + δu T tds S (2) where δ denotes a virtual quantity, ε is the strain tensor and the superscript T specifies the transpose. For space discretization, the displacement field u is approximated in terms of nodal shape functions N and nodal displacements a. Then displacement and strain can be written as u = Na (3) ε = Ba B = LN (4) where B is the strain displacement matrix, L is the linear differential operator. Substituting Eq. (3) and Eq. (4) into Eq. (2) gives or in which δa T Ma = δa T F ext F int (5) Ma = F ext F int (6) M = ρn T Nd F ext = ρn T gd + N T tds S (7) (8) Eq. (6) is identical in the context of FEM and MPM. Since in MPM particles can move through the background mesh, the degrees of freedom and consequently the dimensions of the vectors and matrices presented in Eq. (6) is changed from one time step to another. To increase the computational efficiency lumped mass matrix is used instead of the consistent one m m M L = 2 0 (10) 0 0 m n where n is the number of degrees of freedom. In MPM particle based integration is done i.e. m i = m p N i p (11) where is the total number of particles, m p is the mass of particle p and N i p is the shape function of node i evaluated at particle p. The drawback of using a lumped mass matrix is a slight dissipation in the kinetic energy (Burgess et al. [3]). Using particles to discretize the material (continuum), is done by approximating the density field with the help of the Dirac delta function (Coetzee et al. [4]) ρ (xi ) = m p δ x i x i p (12) where x i is an arbitrary position vector, x i p is the position vector at particle p and the Dirac delta function is defined as follows: δ(x a) = 0 x a x = a δ(x a)dx = 1 and (13) F int = B T σd (9) By considering these points into account, Eq. (8) and Eq. (9) can be rewritten as:

4 F int = B P T σ P P F ext = N P i m P g + N P i t P (14) (15) ector of nodal displacements is calculated by using an Euler backward integration method. Then the position of particles are updated a t+ t = ta t+ t X P t+ t = X P t + N P a t+ t (20). where P is the volume associated to particle P and t P is the traction force vector from the boundary mapped to the boundary particle P. 3. Explicit time integration scheme Most of the MPM applications are based on the explicit integration scheme. This method is conditionally stable and the time step size should be smaller than the critical time step value dictated by the CFL condition t cr = h min cd (16) where h min is the minimum representative distance in an element and c d is the compression wave speed. Often t is chosen as t = α c t cr (17) where α c is called the Courant s number and is a value between one and zero. Explicit time integration of discretized momentum equation is done via applying an Euler forward integration scheme a t = [M t L ] 1 F t, a p t+ t = a p t + tn p a t (18) t+ t and where t is the time increment, a p a p t are the particle velocities at time t and t t + t respectively and a is the nodal accelerations at time t. t+ t The nodal velocities a at time t + t are then calculated from the updated particles velocities, solving the following equation M L t a t+ t m p N p T a p p=1 t+ t (19) 4. Implicit time integration scheme In low frequency motion applications, one can use implicit integration schemes instead of explicit one. Within the unconditionally stable implicit methods bigger time steps can be used so the computational efficiency can improve. Implicit integration scheme used here is based on the method suggested by Stolle et al. [14]. This method is based on a meshless explicit technique and is originated from the work of Nithiarasu [12]. This method is used together with the transient or dynamic relaxation to reach the quasi static solution. The formulation for the transient relaxation is as follows: M TR (a 1 n+1 a 1 n ) = R 1 n D 1 n + M a 1 n a 0 t (21) a 1 n+1 = a 1 n + a 1 n+1 n a 0 t (22) where n is the iteration counter, subscripts correspond to the time stepping stage, M TR is the transient mass matrix, R is the residual force vector and D is the damping force vector based on the suggestion by FLAC [7] which considers the damping force as a coefficient multiplied by the out of balance force in opposite direction of the velocity i.e. D = βr/ where β is a damping coefficient (β 0.7 approximately corresponds to critical damping). M TR is calculated from ρ TR ρ TR = ρ DR / t cr (23) ρ DR = c d t 2 cr h min (24)

5 where ρ DR is related to dynamic relaxation. It should be noticed that M TR is a property of the mesh and does not depend on the particles. See Sauve and Metzger [13] for more information on the relaxation techniques. The idea of the implicit formulation, Eq. (21) and Eq. (22), is to update the displacement and velocity vector until the right hand side of Eq. (21) becomes smaller than a tolerance value. is 640 with 10 particles per element. Plainstrain condition is assumed with roller boundaries along the left side of the mesh. The deformation of the slope was obtained by increasing the unit weight of the soil to γ = 80 kn m 3. Total displacement of the particles, accumulated shear strains, vertical stresses and displacement of a particle initially located at the crest of the slope using explicit integration scheme can be seen in Figs. 3-6 respectively. 5. Remedy to overcome limitations of low order elements In order to overcome the locking phenomena appears in low order elements (here linear triangular elements), the method suggested by Detournay & Dzik [6] is applied. This method is based on the nodal volumetric strain averaging. First the strain is decomposed into the volumetric and deviatoric parts. Then nodal volumetric strains are computed based on the average volumetric strain values of the elements share the node. At last the volumetric strain of the element is the average volumetric strain values of the nodes of that element. For further information on the method refer to Detournay & Dzik [6]. Fig. 2 Mesh of the slope problem 6. Numerical example: Slope stability One of the benchmark problems in large deformation analysis in the field of geomechanics is the slope stability problem. Here a 2D example of slope stability is analyzed using the explicit integration scheme described above. The slope has a height of 1 m, a base length of 2.1 m and an inclination angle of 45. The soil of the slope is assumed to behave according to the elastic-perfectly plastic Mohr-Coulomb model with modulus of elasticity E = 200 kn/m 2, a Poisson s ratio of ν = 0.33, a cohesion of c = 1 kn/m 2 and a friction angle of φ = 25. Moreover, a local damping of α = 0.75 is adopted. A time step size of 10 4 s was used during the simulations. The mesh of the problem is shown in Fig. 2. The number of triangular elements in the initial configuration Fig. 3 Total displacement of the particles [m] Fig. 4 Accumulated shear strain of the particles The method suggested by Detournay & Dzik [6] to overcome the locking problem and the implicit implementation of MPM based on the described strategy is currently developed at the Institute of Geotechnics (IGS) of University of Stuttgart.

6 Fig. 5 ertical stress of the particles [N m 2 ] Displacement (m) 0,50 0,40 0,30 0,20 0,10 0,00 Fig. 6 Displacement of a particle initially located at the crest of the slope 7. Summery In this study the governing equations of MPM was described with their spatial discretization. Explicit and Implicit time integration schemes for MPM were formulated. A slope stability problem as a benchmark problem in large deformation analysis in the field of geomechanics was analyzed using explicit time integration scheme. As can be seen MPM is able to simulate large deformatioroblems suitably. The results show a good analogy to the work of other researchers like [10]. The implicit implementation based on the suggested method is currently developed at the Institute of Geotechnics (IGS) of University of Stuttgart. References Displacement of a particle initially located at the crest of the slope 0,0 2,5 5,0 7,4 9,9 12,4 14,9 17,4 19,9 22,3 24,8 27,3 29,8 32,3 34,8 Time (s) [1] Bardenhagen, S.G., Brackbill, J.U. & Sulsky, D., The material-point method for granular materials, Computer Methods in Applied Mechanics and Engineering, ol. 187, , [2] Beuth, L., Benz, T., ermeer, P.A., Coetzee, C.J., Bonnier, P. & an Den Berg, P., Formulation and validation of a quasistatic material point method, In Proceedings of the 10 th International Symposium on Numerical Methods in Geomechanics, ol. 10, , [3] Burgess, D., Sulsky, D. & Brackbill, J.U., Mass matrix formulation of the FLIP particlein-cell method, Journal of Computational Physics, 103, 1-15, [4] Coetzee, C.J., ermeer, P.A. & Basson, A.H., The modeling of anchors using the material point method, International Journal for Numerical and Analytical Methods in Geomechanics, 29, , [5] Cummins, S.J. & Brackbill, J.U., An implicit particle-in-cell method for granular materials, Journal of Computational Physics, 180, , [6] Detournay, C. & Dzik, E., Nodal mixed discretization for tetrahedral elements, 4 th International FLAC Symposium on Numerical Modeling in Geomechanics, Inc. Paper No , [7] FLAC, Fast Lagrangian Analysis of Continua: Theory and Background, Itasca Consulting Group, Minnesota, USA, [8] Guilkey, J.E. & Weiss, J.A., Implicit time integration for the material point method: Quantitative and algorithmic comparisons with the finite element method, International Journal for Numerical Methods in Engineering, 57, , [9] Harlow, F.H., The particle-in-cell computing method in fluid dynamics, Methods in Computational Physic, 3, , [10] Jassim, I.K., Formulation of a Dynamic Material Point Method (MPM) for Geomechanical Problems, PhD thesis, Institute of Geotechnical Engineering, University of Stuttgart, [11] Jassim, I.K., Hamad, F.M. & ermeer P.A., Dynamic material point method with applications in geomechanics, Proceeding of the 2nd International Symposium on Computational Geomechanics, Croatia, April [12] Nithiarasu, P., A matrix free fractional step method for static and dynamic incompressible solid mechanics,

7 International Journal for Computational Methods in Engineering Science and Mechanics, 7, , [13] Sauve, R.G. & Metzger, D.R., Advances in dynamic relaxation techniques for nonlinear finite element analysis, Journal of Pressure essel Technology, 117, , [14] Stolle, D., Jassim, I. & ermeer, P., Accurate simulation of incompressible problems in geomechanics, Computer Methods in Mechanics, , [15] Sulsky, D. & Kaul, A., Implicit dynamics in the material-point method, Computer Methods in Applied Mechanics and Engineering, 193, , [16] Sulsky, D. & Schreyer, H.L., Axisymmetric form of the material point method with applications to upsetting and Taylor impact problems, Computer Methods in Applied Mechanics and Engineering, ol. 139, , [17] Sulsky, D., Zhou, S.J. & Schreyer, H.L. Application of a particle-in-cell method to solid mechanics, Computer Physics Communications, ol. 87, , [18] Więckowski, Z., Analysis of granular flow by the material point method, European Conference on Computational Mechanics, Cracow, Poland, June 2001.